Different (equivalent?) definition of Axiom schema of replacement

Question

I recently started uni and, as almost everyone, immediately got confused by the axiom schema of replacement. I have searched the web to find some interpretations of it but none of the definitions I came across so far matches one given to me by my professor. He told me that the lectures are based on A course on set theory by Ernest Schimmerling, however, I found that the axiom of replacement is written there same as in the other places I looked, for example on this post on MSE, this post on MSE and this site in Polish.

The definition we were given is close but differs slightly in the use of quantifiers compared to the last link. Translated to english, our formulation is as follows:

(A2$\phi$) Let $\phi(u,v)$ be a formula and $z$ be bound. The axiom of replacement is as follws:

$\mathop{\forall}\limits_{u}\mathop{\forall}\limits_{v}\mathop{\forall}\limits_{w}(\phi(u,v) \wedge\phi(u,w)\Longrightarrow v=w)\Longrightarrow\mathop{\forall}\limits_{a}\mathop{\exists}\limits_{z}\mathop{\forall}\limits_{v}(v\in z \Longleftrightarrow\mathop{\exists}\limits_{u\in a}\phi(u,v))$

While on the Polish site, which has the closest definition to this, there is a slight difference, as it is written as:

Axiom of replacement For any formula $\phi$ not containing any free variables other than $w$ and $v$ the following formula holds:

$\mathop{\forall}\limits_{w}\mathop{\exists}\limits_{u}\mathop{\forall}\limits_{v}(\phi \Longrightarrow u=v)\Longrightarrow\mathop{\forall}\limits_{x}\mathop{\exists}\limits_{y}\mathop{\forall}\limits_{v}(v\in y \Longleftrightarrow\mathop{\exists}\limits_{w\in x}\phi)$

I've tried to tie these two definitions together, but I lack expertiese to see that they are equivalent, expecially since I was not able to find our fourmulation anywhere on the web.

So ultimately I'm asking — are these two definitions in fact equivalent or was a quantifier misplaced? And, do you know any resources that take the first definition?

Thanks for your help.

Answer 1

You should parse both of these formulas (I'll follow the variable namings of the first one) as:

If $\phi(u,v)$ is a class function i.e. for every set $u$ there is at most one $v$ such that $\phi(u,v)$ holds, then for any set $a$, there is a set $z$ such that $z$ is equal to the range of $\phi\upharpoonright a.$ (In other words, if a class function's domain is a set, then its range is a set.)

They just differ in how they express the first part, that $\phi$ is a class function. In the first case, they say that if $\phi(u,v)$ and $\phi(u,w)$ both hold, then $v=w$ (i.e. there is at most one value for the function wherever defined). In the second case they say that for any $u$ there is a $v$ such that if $\phi(u,w)$ holds, then $w=v,$ which amounts to the same thing.